Microscopic theory of the nuclear collective model

Abstract

This article reviews the development of a microscopic theory of nuclear collective structure as a submodel of the nuclear-shell model. It starts by showing how the so-called geometrical (Bohr-Mottelson-Frankfurt) collective model must be augmented by the addition of vortex spin degrees of freedom to make it compatible with the shell model. A unified symplectic model emerges that can be applied both with phenomenological and microscopic interactions. Examples are given of both kinds of calculation. It is shown how the full shell model space can be expressed in an Sp(3, R ) 2 U(3) basis in which form it naturally factors into collective and intrinsic subspaces. In this way, the collective content of a shell model state becomes immediately apparent. Thus a shell model interpretation is given of collective states, including the low-lying rotational bands, the so-called beta and gamma vibrations and the giant monopole and quadrupole resonances.

Symmetry-guided large-scale shell-model theory

“…[43]). That SU(3) plays a key role tracks with the seminal work of Elliott [39,40], and is further reinforced by the fact that SU (3) underpins the microscopic symplectic model [41,42], which provides a theoretical framework for understanding deformation-dominated collective phenomena in atomic nuclei [42,44] and which naturally contains low-energy shape coexisting excitations [35]. We also discuss complementary symmetries that provide alternative reorganization (classification) of the model space.…”

“…While this model was not microscopic, it discussed spatial degrees of the combined many-particle system (spatial deformation and rotations of "shapes"), which suggested a relevant LS-coupling scheme, with single-particle states labeled by ηlm l sm s t z , for which, e.g., a two-particle basis state looks like, {a † ηlstz × a † η l s t z } (LS)JM |0 , where a † is the usual particle creation operator. Indeed, the microscopic Elliott model [39,40,66] and its multi-shell expansion, the symplectic shell model [41,42] that provides a microscopic formulation of the Bohr-Mottelson collective model, have soon after confirmed the relevance of the LS-coupling scheme, while providing a unique and physically relevant organization of the shell-model space, as discussed in Secs. 3.2 and 3.3.…”

“…We focus on the symplectic Sp(3, R) symmetry and its deformation-related SU(3) subgroup, which underpins the Elliott model [39,40], the symplectic model [41,42] and the ab initio SA-NCSM [20], and review the broad scope of nuclei, where these symmetries have been found to play a key role -from the lightest p-shell systems of 6 Li and 8 Be, through sd-shell intermediate-mass nuclei, up to strongly deformed nuclei of the rare-earth and actinide regions (see also the review Ref. [43]).…”

Shape isomers and clusterization in the28Si nucleus

“…The SU(3) quantum numbers uniquely determine the quadrupole shape of the nucleus [13], and so can be used to determine the properties of shape isomers, by carrying out a self-consistent calculation with respect to the quadrupole shape of nucleus [14,15]. In practice, this means the continuous variation of the quadrupole deformation (β in , γ in ), as an input for the Nilsson-model, and determining the effective U(3) quantum numbers or, from them, the corresponding β out , γ out quadrupole deformation.…”

Pseudospin symmetry in nuclear structure and its supersymmetric representation

“…Based on this concept, a simple but useful pseudo-SU(3) model was proposed, and it was generalized to be the pseudo-symplectic model [16,17,18,19]. The concept of pseudospin symmetry has been also widely used in the odd-mass nuclei in the interacting Boson-Fermion model [20].…”